A very similar notion is that of an irreducible morphism.
In an abelian category, a morphism is called a split monomorphism if it is an inclusion of an direct summand, and a split epimorphism if it is the projection onto a direct summand. A morphism is called split if it has either of the above properties.
A morphism $f$ is called irreducible if it is not split but, whenever $f=st$, either $s$ is a split monomorphism or $t$ is a split epimorphism.
I have never seen this definition used outside abelian categories, but I think it makes sense in any category: We can define split monomorphisms as the maps which have right inverses and split epimorphisms as the maps which have left inverses.